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Theorem chfacfscmulgsum 22867
Description: Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
Hypotheses
Ref Expression
chfacfisf.a 𝐴 = (𝑁 Mat 𝑅)
chfacfisf.b 𝐵 = (Base‘𝐴)
chfacfisf.p 𝑃 = (Poly1𝑅)
chfacfisf.y 𝑌 = (𝑁 Mat 𝑃)
chfacfisf.r × = (.r𝑌)
chfacfisf.s = (-g𝑌)
chfacfisf.0 0 = (0g𝑌)
chfacfisf.t 𝑇 = (𝑁 matToPolyMat 𝑅)
chfacfisf.g 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
chfacfscmulcl.x 𝑋 = (var1𝑅)
chfacfscmulcl.m · = ( ·𝑠𝑌)
chfacfscmulcl.e = (.g‘(mulGrp‘𝑃))
chfacfscmulgsum.p + = (+g𝑌)
Assertion
Ref Expression
chfacfscmulgsum (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑌   𝑛,𝑏   𝑛,𝑠,𝐵   0 ,𝑛   𝐵,𝑖,𝑠   𝑖,𝐺   𝑖,𝑀   𝑖,𝑁   𝑅,𝑖   𝑖,𝑋   𝑖,𝑌   ,𝑖   · ,𝑏,𝑖   𝑇,𝑛   ,𝑛   × ,𝑛   𝑖,𝑛
Allowed substitution hints:   𝐴(𝑖,𝑛,𝑠,𝑏)   𝐵(𝑏)   𝑃(𝑖,𝑛,𝑠,𝑏)   + (𝑖,𝑛,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑖,𝑠,𝑏)   · (𝑛,𝑠)   × (𝑖,𝑠,𝑏)   (𝑛,𝑠,𝑏)   𝐺(𝑛,𝑠,𝑏)   𝑀(𝑠,𝑏)   (𝑖,𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑋(𝑛,𝑠,𝑏)   𝑌(𝑠,𝑏)   0 (𝑖,𝑠,𝑏)

Proof of Theorem chfacfscmulgsum
StepHypRef Expression
1 eqid 2736 . . 3 (Base‘𝑌) = (Base‘𝑌)
2 chfacfisf.0 . . 3 0 = (0g𝑌)
3 chfacfscmulgsum.p . . 3 + = (+g𝑌)
4 crngring 20243 . . . . . . . 8 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
54anim2i 617 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
653adant3 1132 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
7 chfacfisf.p . . . . . . 7 𝑃 = (Poly1𝑅)
8 chfacfisf.y . . . . . . 7 𝑌 = (𝑁 Mat 𝑃)
97, 8pmatring 22699 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ Ring)
106, 9syl 17 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
11 ringcmn 20280 . . . . 5 (𝑌 ∈ Ring → 𝑌 ∈ CMnd)
1210, 11syl 17 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ CMnd)
1312adantr 480 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ CMnd)
14 nn0ex 12534 . . . 4 0 ∈ V
1514a1i 11 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ℕ0 ∈ V)
16 simpll 766 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
17 simplr 768 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))))
18 simpr 484 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
1916, 17, 183jca 1128 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
20 chfacfisf.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
21 chfacfisf.b . . . . 5 𝐵 = (Base‘𝐴)
22 chfacfisf.r . . . . 5 × = (.r𝑌)
23 chfacfisf.s . . . . 5 = (-g𝑌)
24 chfacfisf.t . . . . 5 𝑇 = (𝑁 matToPolyMat 𝑅)
25 chfacfisf.g . . . . 5 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))
26 chfacfscmulcl.x . . . . 5 𝑋 = (var1𝑅)
27 chfacfscmulcl.m . . . . 5 · = ( ·𝑠𝑌)
28 chfacfscmulcl.e . . . . 5 = (.g‘(mulGrp‘𝑃))
2920, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulcl 22864 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
3019, 29syl 17 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ ℕ0) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
3120, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulfsupp 22866 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖))) finSupp 0 )
32 nn0disj 13685 . . . 4 ((0...(𝑠 + 1)) ∩ (ℤ‘((𝑠 + 1) + 1))) = ∅
3332a1i 11 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0...(𝑠 + 1)) ∩ (ℤ‘((𝑠 + 1) + 1))) = ∅)
34 nnnn0 12535 . . . . . 6 (𝑠 ∈ ℕ → 𝑠 ∈ ℕ0)
35 peano2nn0 12568 . . . . . 6 (𝑠 ∈ ℕ0 → (𝑠 + 1) ∈ ℕ0)
3634, 35syl 17 . . . . 5 (𝑠 ∈ ℕ → (𝑠 + 1) ∈ ℕ0)
37 nn0split 13684 . . . . 5 ((𝑠 + 1) ∈ ℕ0 → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
3836, 37syl 17 . . . 4 (𝑠 ∈ ℕ → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
3938ad2antrl 728 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ℕ0 = ((0...(𝑠 + 1)) ∪ (ℤ‘((𝑠 + 1) + 1))))
401, 2, 3, 13, 15, 30, 31, 33, 39gsumsplit2 19948 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))))))
41 simpll 766 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
42 simplr 768 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))))
43 nncn 12275 . . . . . . . . . . . . 13 (𝑠 ∈ ℕ → 𝑠 ∈ ℂ)
44 add1p1 12519 . . . . . . . . . . . . 13 (𝑠 ∈ ℂ → ((𝑠 + 1) + 1) = (𝑠 + 2))
4543, 44syl 17 . . . . . . . . . . . 12 (𝑠 ∈ ℕ → ((𝑠 + 1) + 1) = (𝑠 + 2))
4645ad2antrl 728 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑠 + 1) + 1) = (𝑠 + 2))
4746fveq2d 6909 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (ℤ‘((𝑠 + 1) + 1)) = (ℤ‘(𝑠 + 2)))
4847eleq2d 2826 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↔ 𝑖 ∈ (ℤ‘(𝑠 + 2))))
4948biimpa 476 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → 𝑖 ∈ (ℤ‘(𝑠 + 2)))
5020, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmul0 22865 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ (ℤ‘(𝑠 + 2))) → ((𝑖 𝑋) · (𝐺𝑖)) = 0 )
5141, 42, 49, 50syl3anc 1372 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (ℤ‘((𝑠 + 1) + 1))) → ((𝑖 𝑋) · (𝐺𝑖)) = 0 )
5251mpteq2dva 5241 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))) = (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 ))
5352oveq2d 7448 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )))
544, 9sylan2 593 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
55 ringmnd 20241 . . . . . . . . . 10 (𝑌 ∈ Ring → 𝑌 ∈ Mnd)
5654, 55syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Mnd)
57563adant3 1132 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Mnd)
58 fvex 6918 . . . . . . . 8 (ℤ‘((𝑠 + 1) + 1)) ∈ V
5957, 58jctir 520 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V))
6059adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V))
612gsumz 18850 . . . . . 6 ((𝑌 ∈ Mnd ∧ (ℤ‘((𝑠 + 1) + 1)) ∈ V) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
6260, 61syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ 0 )) = 0 )
6353, 62eqtrd 2776 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = 0 )
6463oveq2d 7448 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + 0 ))
65 fzfid 14015 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (0...(𝑠 + 1)) ∈ Fin)
66 elfznn0 13661 . . . . . . . 8 (𝑖 ∈ (0...(𝑠 + 1)) → 𝑖 ∈ ℕ0)
6766, 19sylan2 593 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ ℕ0))
6867, 29syl 17 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...(𝑠 + 1))) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
6968ralrimiva 3145 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑖 ∈ (0...(𝑠 + 1))((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
701, 13, 65, 69gsummptcl 19986 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌))
711, 3, 2mndrid 18769 . . . 4 ((𝑌 ∈ Mnd ∧ (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌)) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
7257, 70, 71syl2an2r 685 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + 0 ) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
7364, 72eqtrd 2776 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ (ℤ‘((𝑠 + 1) + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
7434ad2antrl 728 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℕ0)
751, 3, 13, 74, 68gsummptfzsplit 19951 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖))))))
76 elfznn0 13661 . . . . . . 7 (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0)
7776, 30sylan2 593 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
781, 3, 13, 74, 77gsummptfzsplitl 19952 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖))))))
7957adantr 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Mnd)
80 0nn0 12543 . . . . . . . 8 0 ∈ ℕ0
8180a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 0 ∈ ℕ0)
8220, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulcl 22864 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 0 ∈ ℕ0) → ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
8381, 82mpd3an3 1463 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌))
84 oveq1 7439 . . . . . . . . 9 (𝑖 = 0 → (𝑖 𝑋) = (0 𝑋))
85 fveq2 6905 . . . . . . . . 9 (𝑖 = 0 → (𝐺𝑖) = (𝐺‘0))
8684, 85oveq12d 7450 . . . . . . . 8 (𝑖 = 0 → ((𝑖 𝑋) · (𝐺𝑖)) = ((0 𝑋) · (𝐺‘0)))
871, 86gsumsn 19973 . . . . . . 7 ((𝑌 ∈ Mnd ∧ 0 ∈ ℕ0 ∧ ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((0 𝑋) · (𝐺‘0)))
8879, 81, 83, 87syl3anc 1372 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((0 𝑋) · (𝐺‘0)))
8988oveq2d 7448 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {0} ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))))
9078, 89eqtrd 2776 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))))
91 ovexd 7467 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑠 + 1) ∈ V)
92 1nn0 12544 . . . . . . . 8 1 ∈ ℕ0
9392a1i 11 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 1 ∈ ℕ0)
9474, 93nn0addcld 12593 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
9520, 21, 7, 8, 22, 23, 2, 24, 25, 26, 27, 28chfacfscmulcl 22864 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ (𝑠 + 1) ∈ ℕ0) → (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
9694, 95mpd3an3 1463 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))
97 oveq1 7439 . . . . . . 7 (𝑖 = (𝑠 + 1) → (𝑖 𝑋) = ((𝑠 + 1) 𝑋))
98 fveq2 6905 . . . . . . 7 (𝑖 = (𝑠 + 1) → (𝐺𝑖) = (𝐺‘(𝑠 + 1)))
9997, 98oveq12d 7450 . . . . . 6 (𝑖 = (𝑠 + 1) → ((𝑖 𝑋) · (𝐺𝑖)) = (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))
1001, 99gsumsn 19973 . . . . 5 ((𝑌 ∈ Mnd ∧ (𝑠 + 1) ∈ V ∧ (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))
10179, 91, 96, 100syl3anc 1372 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))
10290, 101oveq12d 7450 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (𝑌 Σg (𝑖 ∈ {(𝑠 + 1)} ↦ ((𝑖 𝑋) · (𝐺𝑖))))) = (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))))
103 fzfid 14015 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (1...𝑠) ∈ Fin)
104 simpll 766 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵))
105 simplr 768 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))))
106 elfznn 13594 . . . . . . . . . 10 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ)
107106nnnn0d 12589 . . . . . . . . 9 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℕ0)
108107adantl 481 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → 𝑖 ∈ ℕ0)
109104, 105, 108, 29syl3anc 1372 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
110109ralrimiva 3145 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ∀𝑖 ∈ (1...𝑠)((𝑖 𝑋) · (𝐺𝑖)) ∈ (Base‘𝑌))
1111, 13, 103, 110gsummptcl 19986 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌))
1121, 3mndass 18757 . . . . 5 ((𝑌 ∈ Mnd ∧ ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) ∈ (Base‘𝑌) ∧ ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))))
11379, 111, 83, 96, 112syl13anc 1373 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))))
114106nnne0d 12317 . . . . . . . . . . . . . 14 (𝑖 ∈ (1...𝑠) → 𝑖 ≠ 0)
115114ad2antlr 727 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ 0)
116 neeq1 3002 . . . . . . . . . . . . . 14 (𝑛 = 𝑖 → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
117116adantl 481 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ 0 ↔ 𝑖 ≠ 0))
118115, 117mpbird 257 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ 0)
119 eqneqall 2950 . . . . . . . . . . . 12 (𝑛 = 0 → (𝑛 ≠ 0 → 0 = (𝑇‘(𝑏‘(𝑖 − 1)))))
120118, 119mpan9 506 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = (𝑇‘(𝑏‘(𝑖 − 1))))
121 simplr 768 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 𝑛 = 𝑖)
122 eqeq1 2740 . . . . . . . . . . . . . . . . 17 (0 = 𝑛 → (0 = 𝑖𝑛 = 𝑖))
123122eqcoms 2744 . . . . . . . . . . . . . . . 16 (𝑛 = 0 → (0 = 𝑖𝑛 = 𝑖))
124123adantl 481 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (0 = 𝑖𝑛 = 𝑖))
125121, 124mpbird 257 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → 0 = 𝑖)
126125fveq2d 6909 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑏‘0) = (𝑏𝑖))
127126fveq2d 6909 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → (𝑇‘(𝑏‘0)) = (𝑇‘(𝑏𝑖)))
128127oveq2d 7448 . . . . . . . . . . 11 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) = ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))
129120, 128oveq12d 7450 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ 𝑛 = 0) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
130 elfz2 13555 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) ↔ ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)))
131 zleltp1 12670 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ ℤ ∧ 𝑠 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
132131ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
1331323adant1 1130 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖𝑠𝑖 < (𝑠 + 1)))
134133biimpcd 249 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖𝑠 → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
135134adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((1 ≤ 𝑖𝑖𝑠) → ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 < (𝑠 + 1)))
136135impcom 407 . . . . . . . . . . . . . . . . . . . 20 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → 𝑖 < (𝑠 + 1))
137136orcd 873 . . . . . . . . . . . . . . . . . . 19 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖))
138 zre 12619 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ ℤ → 𝑠 ∈ ℝ)
139 1red 11263 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑠 ∈ ℤ → 1 ∈ ℝ)
140138, 139readdcld 11291 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 ∈ ℤ → (𝑠 + 1) ∈ ℝ)
141 zre 12619 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
142140, 141anim12ci 614 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
1431423adant1 1130 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ))
144 lttri2 11344 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ ℝ ∧ (𝑠 + 1) ∈ ℝ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
145143, 144syl 17 . . . . . . . . . . . . . . . . . . . 20 ((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
146145adantr 480 . . . . . . . . . . . . . . . . . . 19 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → (𝑖 ≠ (𝑠 + 1) ↔ (𝑖 < (𝑠 + 1) ∨ (𝑠 + 1) < 𝑖)))
147137, 146mpbird 257 . . . . . . . . . . . . . . . . . 18 (((1 ∈ ℤ ∧ 𝑠 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ (1 ≤ 𝑖𝑖𝑠)) → 𝑖 ≠ (𝑠 + 1))
148130, 147sylbi 217 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑠) → 𝑖 ≠ (𝑠 + 1))
149148ad2antlr 727 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 ≠ (𝑠 + 1))
150 neeq1 3002 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑖 → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
151150adantl 481 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 ≠ (𝑠 + 1) ↔ 𝑖 ≠ (𝑠 + 1)))
152149, 151mpbird 257 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ≠ (𝑠 + 1))
153152adantr 480 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → 𝑛 ≠ (𝑠 + 1))
154153neneqd 2944 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → ¬ 𝑛 = (𝑠 + 1))
155154pm2.21d 121 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → (𝑛 = (𝑠 + 1) → (𝑇‘(𝑏𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
156155imp 406 . . . . . . . . . . 11 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ 𝑛 = (𝑠 + 1)) → (𝑇‘(𝑏𝑠)) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
157106nnred 12282 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1...𝑠) → 𝑖 ∈ ℝ)
158 eleq1w 2823 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑖 → (𝑛 ∈ ℝ ↔ 𝑖 ∈ ℝ))
159157, 158syl5ibrcom 247 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) → (𝑛 = 𝑖𝑛 ∈ ℝ))
160159adantl 481 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝑛 = 𝑖𝑛 ∈ ℝ))
161160imp 406 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 ∈ ℝ)
16274nn0red 12590 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℝ)
163162ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑠 ∈ ℝ)
164 1red 11263 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 1 ∈ ℝ)
165163, 164readdcld 11291 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑠 + 1) ∈ ℝ)
166130, 136sylbi 217 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (1...𝑠) → 𝑖 < (𝑠 + 1))
167166ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑖 < (𝑠 + 1))
168 breq1 5145 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑖 → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
169168adantl 481 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → (𝑛 < (𝑠 + 1) ↔ 𝑖 < (𝑠 + 1)))
170167, 169mpbird 257 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → 𝑛 < (𝑠 + 1))
171161, 165, 170ltnsymd 11411 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ¬ (𝑠 + 1) < 𝑛)
172171pm2.21d 121 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → ((𝑠 + 1) < 𝑛0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
173172ad2antrr 726 . . . . . . . . . . . . 13 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → ((𝑠 + 1) < 𝑛0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
174173imp 406 . . . . . . . . . . . 12 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ (𝑠 + 1) < 𝑛) → 0 = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
175 simp-4r 783 . . . . . . . . . . . . . . 15 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → 𝑛 = 𝑖)
176175fvoveq1d 7454 . . . . . . . . . . . . . 14 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏‘(𝑛 − 1)) = (𝑏‘(𝑖 − 1)))
177176fveq2d 6909 . . . . . . . . . . . . 13 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏‘(𝑛 − 1))) = (𝑇‘(𝑏‘(𝑖 − 1))))
178175fveq2d 6909 . . . . . . . . . . . . . . 15 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑏𝑛) = (𝑏𝑖))
179178fveq2d 6909 . . . . . . . . . . . . . 14 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → (𝑇‘(𝑏𝑛)) = (𝑇‘(𝑏𝑖)))
180179oveq2d 7448 . . . . . . . . . . . . 13 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇𝑀) × (𝑇‘(𝑏𝑛))) = ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))
181177, 180oveq12d 7450 . . . . . . . . . . . 12 ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) ∧ ¬ (𝑠 + 1) < 𝑛) → ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
182174, 181ifeqda 4561 . . . . . . . . . . 11 (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) ∧ ¬ 𝑛 = (𝑠 + 1)) → if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
183156, 182ifeqda 4561 . . . . . . . . . 10 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
184129, 183ifeqda 4561 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) ∧ 𝑛 = 𝑖) → if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
185 ovexd 7467 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))) ∈ V)
18625, 184, 108, 185fvmptd2 7023 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → (𝐺𝑖) = ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))
187186oveq2d 7448 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (1...𝑠)) → ((𝑖 𝑋) · (𝐺𝑖)) = ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))
188187mpteq2dva 5241 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖))) = (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))))
189188oveq2d 7448 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = (𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))))
190 nn0p1gt0 12557 . . . . . . . . . . . . . 14 (𝑠 ∈ ℕ0 → 0 < (𝑠 + 1))
191 0red 11265 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ℕ0 → 0 ∈ ℝ)
192 ltne 11359 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
193191, 192sylan 580 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑠 + 1) ≠ 0)
194 neeq1 3002 . . . . . . . . . . . . . . 15 (𝑛 = (𝑠 + 1) → (𝑛 ≠ 0 ↔ (𝑠 + 1) ≠ 0))
195193, 194syl5ibrcom 247 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℕ0 ∧ 0 < (𝑠 + 1)) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
19634, 190, 195syl2anc2 585 . . . . . . . . . . . . 13 (𝑠 ∈ ℕ → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
197196ad2antrl 728 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑛 = (𝑠 + 1) → 𝑛 ≠ 0))
198197imp 406 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → 𝑛 ≠ 0)
199 eqneqall 2950 . . . . . . . . . . 11 (𝑛 = 0 → (𝑛 ≠ 0 → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏𝑠))))
200198, 199mpan9 506 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ 𝑛 = 0) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = (𝑇‘(𝑏𝑠)))
201 iftrue 4530 . . . . . . . . . . 11 (𝑛 = (𝑠 + 1) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = (𝑇‘(𝑏𝑠)))
202201ad2antlr 727 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) ∧ ¬ 𝑛 = 0) → if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))) = (𝑇‘(𝑏𝑠)))
203200, 202ifeqda 4561 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑛 = (𝑠 + 1)) → if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))) = (𝑇‘(𝑏𝑠)))
20474, 35syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑠 + 1) ∈ ℕ0)
205 fvexd 6920 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏𝑠)) ∈ V)
20625, 203, 204, 205fvmptd2 7023 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝐺‘(𝑠 + 1)) = (𝑇‘(𝑏𝑠)))
207206oveq2d 7448 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) = (((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))))
20843ad2ant2 1134 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
209 eqid 2736 . . . . . . . . . . . . . 14 (Base‘𝑃) = (Base‘𝑃)
21026, 7, 209vr1cl 22220 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
211208, 210syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑋 ∈ (Base‘𝑃))
212 eqid 2736 . . . . . . . . . . . . . 14 (mulGrp‘𝑃) = (mulGrp‘𝑃)
213212, 209mgpbas 20143 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
214 eqid 2736 . . . . . . . . . . . . . 14 (1r𝑃) = (1r𝑃)
215212, 214ringidval 20181 . . . . . . . . . . . . 13 (1r𝑃) = (0g‘(mulGrp‘𝑃))
216213, 215, 28mulg0 19093 . . . . . . . . . . . 12 (𝑋 ∈ (Base‘𝑃) → (0 𝑋) = (1r𝑃))
217211, 216syl 17 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0 𝑋) = (1r𝑃))
2187ply1crng 22201 . . . . . . . . . . . . . . 15 (𝑅 ∈ CRing → 𝑃 ∈ CRing)
219218anim2i 617 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
2202193adant3 1132 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
2218matsca2 22427 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
222220, 221syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 = (Scalar‘𝑌))
223222fveq2d 6909 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (1r𝑃) = (1r‘(Scalar‘𝑌)))
224217, 223eqtrd 2776 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (0 𝑋) = (1r‘(Scalar‘𝑌)))
225224adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (0 𝑋) = (1r‘(Scalar‘𝑌)))
226225oveq1d 7447 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0 𝑋) · (𝐺‘0)) = ((1r‘(Scalar‘𝑌)) · (𝐺‘0)))
2277, 8pmatlmod 22700 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑌 ∈ LMod)
2284, 227sylan2 593 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
2292283adant3 1132 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ LMod)
23020, 21, 7, 8, 22, 23, 2, 24, 25chfacfisf 22861 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
2314, 230syl3anl2 1414 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
232231, 81ffvelcdmd 7104 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝐺‘0) ∈ (Base‘𝑌))
233 eqid 2736 . . . . . . . . . 10 (Scalar‘𝑌) = (Scalar‘𝑌)
234 eqid 2736 . . . . . . . . . 10 (1r‘(Scalar‘𝑌)) = (1r‘(Scalar‘𝑌))
2351, 233, 27, 234lmodvs1 20889 . . . . . . . . 9 ((𝑌 ∈ LMod ∧ (𝐺‘0) ∈ (Base‘𝑌)) → ((1r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
236229, 232, 235syl2an2r 685 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((1r‘(Scalar‘𝑌)) · (𝐺‘0)) = (𝐺‘0))
237 iftrue 4530 . . . . . . . . 9 (𝑛 = 0 → if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
238 ovexd 7467 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) ∈ V)
23925, 237, 81, 238fvmptd3 7038 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝐺‘0) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
240226, 236, 2393eqtrd 2780 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((0 𝑋) · (𝐺‘0)) = ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
241207, 240oveq12d 7450 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 𝑋) · (𝐺‘0))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
2421, 3cmncom 19817 . . . . . . 7 ((𝑌 ∈ CMnd ∧ ((0 𝑋) · (𝐺‘0)) ∈ (Base‘𝑌) ∧ (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) ∈ (Base‘𝑌)) → (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 𝑋) · (𝐺‘0))))
24313, 83, 96, 242syl3anc 1372 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))) + ((0 𝑋) · (𝐺‘0))))
244 ringgrp 20236 . . . . . . . . 9 (𝑌 ∈ Ring → 𝑌 ∈ Grp)
24510, 244syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Grp)
246245adantr 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Grp)
247207, 96eqeltrrd 2841 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌))
24810adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Ring)
24924, 20, 21, 7, 8mat2pmatbas 22733 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
2504, 249syl3an2 1164 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝑌))
251250adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇𝑀) ∈ (Base‘𝑌))
252 simpl1 1191 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑁 ∈ Fin)
253208adantr 480 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑅 ∈ Ring)
254 elmapi 8890 . . . . . . . . . . . 12 (𝑏 ∈ (𝐵m (0...𝑠)) → 𝑏:(0...𝑠)⟶𝐵)
255254adantl 481 . . . . . . . . . . 11 ((𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → 𝑏:(0...𝑠)⟶𝐵)
256255adantl 481 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑏:(0...𝑠)⟶𝐵)
257 0elfz 13665 . . . . . . . . . . . 12 (𝑠 ∈ ℕ0 → 0 ∈ (0...𝑠))
25834, 257syl 17 . . . . . . . . . . 11 (𝑠 ∈ ℕ → 0 ∈ (0...𝑠))
259258ad2antrl 728 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → 0 ∈ (0...𝑠))
260256, 259ffvelcdmd 7104 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑏‘0) ∈ 𝐵)
26124, 20, 21, 7, 8mat2pmatbas 22733 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑏‘0) ∈ 𝐵) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
262252, 253, 260, 261syl3anc 1372 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌))
2631, 22ringcl 20248 . . . . . . . 8 ((𝑌 ∈ Ring ∧ (𝑇𝑀) ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏‘0)) ∈ (Base‘𝑌)) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
264248, 251, 262, 263syl3anc 1372 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌))
2651, 2, 23, 3grpsubadd0sub 19046 . . . . . . 7 ((𝑌 ∈ Grp ∧ (((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ∈ (Base‘𝑌) ∧ ((𝑇𝑀) × (𝑇‘(𝑏‘0))) ∈ (Base‘𝑌)) → ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
266246, 247, 264, 265syl3anc 1372 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) + ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
267241, 243, 2663eqtr4d 2786 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0)))))
268189, 267oveq12d 7450 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + (((0 𝑋) · (𝐺‘0)) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
269113, 268eqtrd 2776 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) + ((0 𝑋) · (𝐺‘0))) + (((𝑠 + 1) 𝑋) · (𝐺‘(𝑠 + 1)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
27075, 102, 2693eqtrd 2780 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...(𝑠 + 1)) ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
27140, 73, 2703eqtrd 2780 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wne 2939  Vcvv 3479  cun 3948  cin 3949  c0 4332  ifcif 4524  {csn 4625   class class class wbr 5142  cmpt 5224  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867  Fincfn 8986  cc 11154  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296  cle 11297  cmin 11493  cn 12267  2c2 12322  0cn0 12528  cz 12615  cuz 12879  ...cfz 13548  Basecbs 17248  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18748  Grpcgrp 18952  -gcsg 18954  .gcmg 19086  CMndccmn 19799  mulGrpcmgp 20138  1rcur 20179  Ringcrg 20231  CRingccrg 20232  LModclmod 20859  var1cv1 22178  Poly1cpl1 22179   Mat cmat 22412   matToPolyMat cmat2pmat 22711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-ofr 7699  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-subrng 20547  df-subrg 20571  df-lmod 20861  df-lss 20931  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-ascl 21876  df-psr 21930  df-mvr 21931  df-mpl 21932  df-opsr 21934  df-psr1 22182  df-vr1 22183  df-ply1 22184  df-mamu 22396  df-mat 22413  df-mat2pmat 22714
This theorem is referenced by:  cpmadugsum  22885
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